Logarithm formula proof

Question

Prove:

$$x^{\log(y)}=y^{\log(x)}$$

I have been trying this for the past 1 hour, still cant prove it.

I started with

$$\log_b(y)=m$$ $$\log_b(x)=n$$

To show: $$x^m = y^n$$

How do i proceed? :

Answer 1

Since we can take logs on both sides because taking the log preserves the equality i.e. $$ \log a = \log b \iff a = b $$ in short, we can still prove the statement if are working the logarithm of the equality.

Anyway,

$$ \log \left(x^{\log y}\right) = \log (y) \log (x) $$ similarly $$ \log \left(y^{\log x}\right) = \log (x) \log (y) $$ here I used $\log a^b = b \log a$.

Answer 2

Hint: $\large x^a=\left(e^{\log x}\right)^a=e^{(\log x)\cdot a}$

Answer 3

Start with the following equality:

$$\log(x)\log(y)=\log(x)\log(y)\\ \implies e^{\log(x)\log(y)}=e^{\log(x)\log(y)}\\ \implies \left(e^{\log x}\right)^{\log y}=\left(e^{\log y}\right)^{\log x}\\ \implies x^{\log y}=y^{\log x}$$

Identities used:

• $e^{\log x}=x$
• $(a^m)^n=(a^n)^m=a^{mn}$